Academic journal article Cosmos and History: The Journal of Natural and Social Philosophy

L.E.J. Brouwer and Karl Popper: Two Perspectives on Mathematics

Academic journal article Cosmos and History: The Journal of Natural and Social Philosophy

L.E.J. Brouwer and Karl Popper: Two Perspectives on Mathematics

Article excerpt

INTRODUCTION

This article provides an appraisal of Popper's criticism of L. E. J. Brouwer's intuitionist mathematics. (1) Despite the extensive scholarship on Popper, his engagement with Brouwer's thought has largely been overlooked. Through his critical engagement with Brouwer, Popper provides an eloquent overview of some of the innovative features of his own later objectivist evolutionary epistemology. A direct critique of Brouwer's thought presented in Popper's lecture "Epistemology Without A Knowing Subject" (1967) published in Objective Knowledge: An Evolutionary Approach (1972), however, influence of his engagement with Brouwer on the problem of intuitionism is recurrent in his latter writings. Both Popper and Brouwer agreed that the problem of sequential or rational reasoning and its relation to our immediate perceptions is a crucial problem for human knowledge. For Popper, language or sequential rationality distorts our unconscious immediate perceptions, which comprise the vast bulk of our knowledge at any given time. Brouwer on the other hand, viewed a greater private accessibility of such intuitions or immediate perceptions. For Brouwer, the intuitional ground of mathematics completely separates mathematics from mathematical language. Intuitionistic mathematics is an essentially languageless activity of the mind having its origin in the perception of a move of time. The mathematician is prioritised as the ultimate source of authority over the formalised representation of mathematics. In this context, language, including mathematical language is incapable of providing the means of communicating them to others. Popper, while appreciating the important role of intuitions in identifying problems and deriving solutions was highly critical of Brouwer's subjectivist orientation.

For Popper, the problem of intuition was re-construed in terms of evolutionary cognition as the problem of "unconscious expectations" or "background knowledge". The way this relates to our discursive knowledge is not restricted to mathematics or the hard sciences but is crucial in understanding all instances of human problem solving. This background knowledge does not derive from some pristine source of truth in the subject, but is the result of previous problem solving attempts, which become built into our cognitive apparatus and unconsciously inform our actions in the form of conjectures. What is crucial for Popper, is the way we externalise our subjective knowledge in the form of conjectures, which both enables such knowledge to be criticised, as well as potentially lay bare hitherto unseen implications. For Popper, this was crucial to the way knowledge grows, and is necessary for the development of the self, which is dependent upon linguistic communication. Section 2 provides a brief overview of Brower's intuitionist mathematics relevant to the current discussion. Section 3 then provides a close reading of Popper's analysis and response to Brouwer.

BACKGROUND

Brouwer's intuitionism is related to conceptualism, which holds that abstract entities exist only insofar as they are constructed by the human mind. Thus, for the intuitionist mathematician the abstract entities, which occur in mathematics, such as sequences or order-relations, are all mental constructions. As a result, the amount of abstract entities for an intuitionist mathematician is greatly restricted in comparison to classical mathematics as well as in logicism. Popper did not agree with mathematical logicism, rather he agreed with the "realist" or Platonist doctrine that abstract entities have an existence independent of the human mind. This view is justified on the basis that mathematics is full of abstract entities such as numbers, functions and sets to name a few, which according to Plato exist outside the mind. From this we get the medieval philosophical doctrine of realism held that the mind discovers such entities but does not create them. …

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